### Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

### Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

# Poison, Antidote, Water

##### Stage: 4 and 5 Challenge Level:

Let's play a game ....

I have a large set of cards which are printed 'Poison', 'Antidote' and 'Water'.

I shuffle up the cards and lay out a row of 5 cards, face down.

You choose any two cards which are next to each other and turn them over. Two Poison cards is still Poison, so replace these two cards with a single Poison card. The same goes for two Antidote or Water cards - replace two of these with a single Antidote or Water card respectively. Mixing Water with Poison still leaves Poison, and mixing water with Antidote still leaves Antidote (we ignore the chemistry of concentrations in this game!)

The game is to repeat this procedure until you are left with a single card, and you lose if you end up with Poison.

My question is this: Does the order in which you decide to turn over the cards Sometimes, Always or Never matter? Give examples or a clear argument to illustrate your conclusion. If the cards are totally random, what is the chance of me being Poisoned?

You can play with this interactivity if you wish:

If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.

Hopefully you survived the game of Poison, Antidote Water! Now let's play a similar game based on the classic Scissors, Paper, Stone. In this game Scissors beats Paper, Paper beats Stone and Stone beats Scissors.

I lay out face down a row of 5 cards printed with one of these symbols and you turn over two cards next to each other, replacing it with the winner of the two cards if the two cards are different or removing one of the two cards if they are the same.

My question is again this: Does the order in which you decide to turn over the cards Sometimes, Always or Never matter? Give examples or a clear argument to illustrate your conclusion.

Now consider this: What are the structural similarities and differences between the two games?

And finally: Imagine that you are given a set of cards with 3 different images on them (you choose!). Is it possible to invent a different set of rules such that the order in which the cards are turned over does not affect the result of the game?

Extension: Repeat for a game with sets of cards with 4 images on them.